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We introduce a novel location-based moving mesh algorithm MMSISL in which the arrival points in the Semi-Implicit Semi-Lagrangian (SISL) algorithm are located by using an equidistribution strategy. This algorithm gives a natural coupling between moving mesh methods and SISL methods. It involves little extra cost in implementation as it exploits the interpolation methods already embedded in the SISL algorithm. We apply this method to a number of partial differential equation problems in one-dimension, each of which have sharply defined features. We show that using MMSISL leads to a markedly improved performance over fixed mesh methods, with significantly reduced errors. We also show that unlike many adaptive schemes, no issues arise in the MMSISL algorithm from a CFL condition imposed restriction on the time step.

In this article we survey r-adaptive (or moving grid) methods for solving time-dependent partial differential equations (PDEs). Although these methods have received much less attention than their h- and p-adaptive counterparts, particularly within the finite element community, we review the substantial progress that has been made in developing more robust and reliable algorithms and in understanding the basic principles behind these methods, and we give some numerical examples illustrative of the wide classes of problems for which these methods are suitable alternatives to the traditional ones.More specifically, we first examine the basic geometric properties of moving meshes in both one and higher spatial dimensions, and discuss the discretization process for PDEs on such moving meshes (both structured and unstructured). In particular, we consider the issues of mesh regularity, equidistribution, alignment, and associated variational methods. An overview is given of the general interpolation error analysis for a function or a truncation error on such an adaptive mesh. Guided by these principles, we show how to design effective moving mesh strategies. We then examine in more detail how these strategies can be implemented in practice. The first class of methods which we consider are based upon controlling mesh density and hence are called position-based methods. These make use of a so-called moving mesh PDE (MMPDE) approach and variational methods, as well as optimal transport methods. This is followed by an analysis of methods which have a more Lagrange-like interpretation, and due to this focus are called velocity-based methods. These include the moving finite element method (MFE), the geometric conservation law (GCL) methods, and the deformation map method. Finally, we present a number of specific types of examples for which the use of a moving mesh method is particularly effective in applications. These include scale-invariant problems, blow-up problems, problems with moving fronts and problems in meteorology. We conclude that, whilst r-adaptive methods are still in their relatively early stages of development, with many outstanding questions remaining, they have enormous potential and indeed can produce an optimal form of adaptivity for many problems.

This article considers a new method for generating a moving mesh which is suitable for the numerical solution of partial differential equations (PDEs) in several spatial dimensions. The mesh is obtained by taking the gradient of a (scalar) mesh potential function which satisfies an appropriate nonlinear parabolic partial differential equation. This method gives a new technique for performing $r$-adaptivity based on ideas from optimal transportation combined with the equidistribution principle applied to a (time-varying) scalar monitor function (used successfully in moving mesh methods in one-dimension). Detailed analysis of this new method is presented in which the convergence, regularity, and stability of the mesh is studied. Additionally, this new method is shown to be straightforward to program and implement, requiring the solution of only one simple scalar time-dependent equation in arbitrary dimension, with adaptivity along the boundaries handled automatically. We discuss three preexisting methods in the context of this work. Examples are presented in which either the monitor function is prescribed in advance, or it is given by the solution of a partial differential equation.

This paper constructs and analyses an adaptive moving mesh scheme for the numerical simulation of singular PDEs in one or more spatial dimensions. The scheme is based on computing a Legendre transformation from a regular to a spatially non-uniform mesh via the solution of a relaxed form of the Monge–Ampere equation. The method is shown to preserve the inherent scaling properties of the PDE and to identify natural computational coordinates. Numerical examples are presented in one and two dimensions which demonstrate the effectiveness of this approach.

We discuss a new approach for the numerical computation of self-similar blow-up solutions of certain nonlinear partial differential equations. These solutions become unbounded in finite time at a single point at which there is a growing and increasingly narrow peak. Our main focus is on a quasilinear parabolic problem in one space dimension, but our approach can also be applied to other problems featuring blow-up solutions. For the model we consider here, the problem of the computation of the self-similar solution profile reduces to a nonlinear, second-order ordinary differential equation on an unbounded domain, which is given in implicit form. We demonstrate that a transformation of the independent variable to the interval [0,1] yields a singular problem which facilitates a stable numerical solution. To this end, we implemented a collocation code which is designed especially for implicit second order problems. This approach is compared with the numerical solution by standard methods from the literature and by well-established numerical solvers for ODEs. It turns out that the new solution method compares favorably with previous approaches in its stability and efficiency. Finally, we comment on the applicability of our method to other classes of nonlinear PDEs with blow-up solutions.

We use a variety of careful numerical and semi-analytical methods to investigate two outstanding conjectures on the solutions of the parametrised semi-linear elliptic equation Δu + λu + u5=0, u > 0, where u is defined to be zero on the boundary of a three dimensional domain. This equation is important in analysis and in studies of combustion and polytropic gases. It is known that there is a value λ0 > 0 such that no solutions exist for λ < λ0. McLeod and Schoen, and Bandle and Flucher have given different estimates for λ0; both of which have been conjectured to be exact. We perform a semi-analytic study of solutions on cylindrical domains and construct numerical approximations on cuboid domains using the finite element method combined with a careful post-processing step to reduce the otherwise significant errors. We compute these estimates for cylindrical and cuboid domains, and show that on these domains the estimates do not agree. We conclude that the conjecture of Bandle and Flucher is false. Our numerical computations on cuboid domains are consistent with McLeod's conjecture being true.

We study the blow-up self-similar solutions of the radially symmetric nonlinear Schrodinger equation (NLS) given by iut+urr+d?1rur+u|u|2, with dimension d>2. These solutions become infinite in a finite time T. By a series of careful numerical computations, partly supported by analytic results, we demonstrate that there is a countably infinite set of blow-up self-similar solutions which satisfy a second order complex ordinary differential equation with an integral constraint. These solutions are characterised by the number of oscillations in their amplitude when d is close to 2. The solutions are computed as functions of d and their behaviour in the critical limit as d?2 is investigated. The stability of these solutions is then studied by solving the NLS by using an adaptive numerical method. This method uses moving mesh partial differential equations and exploits the scaling invariance properties of the underlying equation. We demonstrate that the single-humped self-similar solution is globally stable whereas the multi-humped solutions all appear to be unstable.

This paper examines a synthesis of adaptive mesh methods with the use of symmetry to study a partial differential equation. In particular, it considers methods which admit discrete self–similar solutions, examining the convergence of these to the true self–similar solution as well as their stability. Special attention is given to the nonlinear diffusion equation describing flow in a porous medium.

We consider the finite element solution of the parameterized semilinear elliptic equation $\Delta u + \lambda u + u^{5} = 0, u > 0$, where $u$ is defined in the unit cube and is 0 on the boundary of the cube. This equation is important in analysis, and it is known that there is a value $\lambda_{0} > 0$ such that no solutions exist for $\lambda \lambda_{0}$ and for the form of the spurious numerical solutions which are known to exist when $\lambda < \lambda_{0}$. These estimates are then used to post-process the numerical results to obtain a sharp estimate for $\lambda_{0}$ which agrees with the conjectured value.

Copyright (c) 1997 Elsevier Science B.V. All rights reserved. We consider the Cauchy problem for the nonlinear diffusion equation, u t =(u m u x ) x which is posed on an infinite domain. The PDE and a conservation law are invariant to a Lie group of stretchings which is used to construct the invariant quantities, xt −1/(2+m) and ut 1/(2+m) . Using these invariants as similarity variables the problem is reduced to a second order ODE and then integrated to give the well known similarity solutions. The problem is semi-discretized using the method of lines. The mesh movement is governed by the conservation of mass law, so that the computational domain expands as the solution diffuses. The resulting semi-discretization is a system of ODEs which is invariant to the same Lie group as the PDE and so the mesh X i (t) behaves like the discretized invariant Y i t 1/(2+m) and the solution U i (t) like V i t −1/(2+m) . Furthermore, it is shown that, using these invariants, the same reduction and integration is possible in the semi-discrete case as in the continuous case.

We consider the effectiveness of adaptive finite element methods for finding the finite element solutions of the parametrised semi-linear elliptic equation Δ u + λ u + u 5 = 0, u > 0, where u ϵ C 2 (Ω), for a domain Ω ⊂ R and u = 0 on the boundary of Ω. This equation is important in analysis and it is known that there is a value λ 0 > 0 such that no solutions exist for λ 0 and a singularity forms as λ → λ 0 . Furthermore the linear operator L defined by L φ = Δφ + λφ + 5 u 4 φ has a singular inverse in this limit. We demonstrate that conventional adaptive methods (using both static and dynamic regridding) based on usual error estimates fail to give accurate solutions and indeed admit spurious solutions of the differential equation when λ 0 . This is directly due to the lack of invertibility of the operator L . In contrast we show that error estimates which take this into account can give answers to any prescribed tolerance.

In this paper we consider the numerical solution of PDEs with blow-up for which scaling invariance plays a natural role in describing the underlying solution structures. It is a challenging numerical problem to capture the qualitative behaviour in the blow-up region, and the use of nonuniform meshes is essential. We consider moving mesh methods for which the mesh is determined using so-called moving mesh partial differential equations (MMPDEs). Specifically, the underlying PDE and the MMPDE are solved for the blow-up solution and the computational mesh simultaneously. Motivated by the desire for the MMPDE to preserve the scaling invariance of the underlying problem, we study the effect of different choices of MMPDEs and monitor functions. It is shown that for suitable ones the MMPDE solution evolves towards a (moving) mesh which close to the blow-up point automatically places the mesh points in such a manner that the ignition kernel, which is well known to be a natural coordinate in describing the behaviour of blow-up, approaches a constant as $t\to T$ (the blow-up time). Several numerical examples are given to verify the theory for these MMPDE methods and to illustrate their efficacy.

A reaction-diffusion-convection equation with a nonlocal term is studied; the nonlocal operator acts to conserve the spatial integral of the unknown function as time evolves. The equations are parameterised by µ, and for µ = 1 the equation arises as a similarity solution of the Navier-Stokes equations and the nonlocal term plays the role of pressure. For µ = 0, the equation is a nonlocal reaction-diffusion problem. The aim of the paper is to determine for which values of the parameter µ blow-up occurs and to study its form. In particular, interest is focused on the three cases µ < 1/2, µ > 1/2, and µ → 1. It is observed that, for any 0 ≤ µ ≤ 1/2, nonuniform global blow-up occurs; if 1/2 < µ < 1, then the blow-up is global and uniform, while for µ = 1 (the Navier-Stokes equations) there are exact solutions with initial data of arbitrarily large L_∞, L_2, and H^1 norms that decay to zero. Furthermore, one of these exact solutions is proved to be nonlinearly stable in L_2 for arbitrarily large supremum norm. An understanding of this transition from blow-up behaviour to decay behaviour is achieved by a combination of analysis, asymptotics, and numerical techniques.

A reaction-diffusion equation with a nonlocal term is studied. The nonlocal term acts to conserve the spatial integral of the unknown function as time evolves. Such equations give insight into biological and chemical problems where conservation properties predominate. The aim of the paper is to understand how the conservation property affects the nature of blowup. The equation studied has a trivial steady solution that is proved to be stable. Existence of nontrivial steady solutions is proved, and their instability established numerically. Blowup is proved for sufficiently large initial data by using a comparison principle in Fourier space. The nature of the blowup is investigated by a combination of asymptotic and numerical calculations.

SynopsisWe study the asymptotic behaviour as x →∞ of the solutions of the ordinary differential equation problemThis equation generalises the ordinary differential equation obtained by studying the blow-up of the similarity solutions of the semilinear parabolic partial differential equation vt=vxx = ev. We show that if λ≦1, all solutions of (*) tend to —∞ as rapidly as the function —exp (x2/4) (E- solutions). However, if λ>1, then there also exists a solution which tends to –∞, like 2λlog(x) (L-solutions). Thus, the case λ = 1, for which (*) reduces tothe Kassoy equation, is the borderline between two quite different forms of asymptotic behaviour of the function u(x).

In this paper we consider a model for the electric potentialΦarising from the field due to a coronating electrode. This results in the third-order nonlinear partial differential equation ∇. (∆Φ∇Φ) = 0, which is of mixed type and we study here in two spatial dimensions. We describe a new way of analysing this equation by constructing an orthogonal coordinate system (Φ,Ψ) and recasting the equation in terms ofx,yand ∆Φas functions of (Φ,Ψ). It may be regarded as a synthesis of the method of characteristics for a hyperbolic equation and a hodograph method for Laplace’s equation. This results in an effective algorithm for a numerical solution with finite differences.

On etudie l'equation aux derivees partielles non lineaires d'ordre 3 ⊇•(⊇φΔφ)=0 pour deux dimensions d'espaces. On utilise une methode d'hodographe pour deduire des solutions exactes

We study the blow-up behaviour of two reaction-diffusion problems with a quasilinear degenerate diffusion and a superlinear reaction. We show that in each case the blow-up is self-similar, in contrast to the linear diffusion limit of each in which the diffusion is only approximately self-similar. We then investigate the limit of the self-similar behaviour and describe the transition from a stable manifold blow-up behaviour (for quasilinear diffusion) to a centre manifold one for the linear diffusion.

It is a readily observable fact that many physical and mathematical systems possess a degree of symmetry and that a study of this symmetry may give us valuable insight into their behaviour. It is particularly interesting that symmetric systems exist which possess non-symmetric solutions and where this solution branch arises from a symmetry breaking bifurcation on a branch of symmetric solutions. In this paper we shall study an example of such a system. Namely we shall study the symmetry breaking bifurcations (henceforth denoted as SBB’s) which occur on the radially symmetric solution branches of the following semilinear elliptic equation.